It's been a long while since I worked on a 5/01 controller; the operating system of the 5/01 goes back to the late 1980's and it was almost completely replaced by the 5/03 as soon as that came out in 1993. If I was going to replace the controller I'd use the 8K version of the 5/03 rather than trying to get an actual 5/01.
The key symptom that I see is that the controller seems to be reporting that the user program has been erased; you mentioned that it reports the online program name to be "no program".
So, this isn't a conventional fault. It's not a misconfigured I/O slot, or a divide-by-zero, or something else related to the user program. It's a failure that happens when the operating system first boots up on power-up.
If you happen to have an OPC-capable version of RSLinx you can link the S:6 status word to Excel or the OPC Test Client or the Data Table Monitor to check if there's a useful value in there.
I've seen power-cycle problems related to the power supply (sometimes it's the 115/230V switch) and the chassis and the CPU. Which one you change out depends on what you've got for spares and testing. If it were me, I've got a bunch of spare 1747 chassis and power suppplies that are known to be good so I'd put the controller into one of those to test it.